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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Nonradial solvability structure of super-diffusive nonlinear parabolic equations
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by Panagiota Daskalopoulos and Manuel del Pino PDF
Trans. Amer. Math. Soc. 354 (2002), 1583-1599 Request permission

Abstract:

We study the solvability of the Cauchy problem for the nonlinear parabolic equation \[ \frac {\partial u}{\partial t} = \mbox {div} (u^{m-1}\nabla u)\] when $m < 0$ in $\textbf {R}^2$, with $u(x,0)= f(x)$ a given nonnegative function. It is known from earlier works of the authors that the asymptotic radial growth $r^{-2/1-m}$, $r=|x|$ for the spherical averages of $f(x)$ is critical for local solvability, in particular ensuring it if $f$ is radially symmetric. We show that if the initial data $f(x)$ behaves in polar coordinates like $r^{-2/1-m} g(\theta )$, for large $r= |x|$ with $g$ nonnegative and $2\pi$-periodic, then the following holds: If $g$ vanishes on some interval of length $l^* = \frac {(m-1)\pi }{2m} >0$, then there is no local solution of the initial value problem. On the other hand, if such an interval does not exist, then the initial value problem is locally solvable and the time of existence can be estimated explicitly.
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Additional Information
  • Panagiota Daskalopoulos
  • Affiliation: Department of Mathematics, University of California at Irvine, Irvine, California 92697
  • MR Author ID: 353551
  • Email: pdaskalo@math.uci.edu
  • Manuel del Pino
  • Affiliation: Departamento de Ingeniería Matemática and Centro de Modelamiento Matemático, Universidad de Chile, Casilla 170 Correo 3, Santiago, Chile
  • MR Author ID: 56185
  • Email: delpino@dim.uchile.cl
  • Received by editor(s): March 25, 1998
  • Published electronically: December 4, 2001
  • Additional Notes: The first author was partially supported by The Sloan Foundation and by NSF/Conicyt-Chile grant INT-9802406
    The second author was partially supported by grants Lineas Complementarias Fondecyt 8000010 and FONDAP
  • © Copyright 2001 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 354 (2002), 1583-1599
  • MSC (1991): Primary 35K15, 35K55, 35K65; Secondary 35J40
  • DOI: https://doi.org/10.1090/S0002-9947-01-02888-4
  • MathSciNet review: 1873019